Let $A$ be a symmetric matrix with real coefficients and let $O_1$ and $O_2$ be two real orthogonal matrices (i.e. $O_x^T O_x = I$).
What can one say about $$ O_1^T A O_1 + O_2^T A O_2 $$ ?
Is it possible to find two matrices $B$ and $C$ such that $$ O_1^T A O_1 + O_2^T A O_2 = B^{-1} A C $$ ?
It is simple to show that in general (unless $O_1^T A O_1$ and $O_2^T A O_2$ commute) $B \neq C$. I am particularly interested to the case when $A$ is a block matrix and the orthogonal matrices are permutation matrices.
This is not always possible. E.g. when $A=\operatorname{diag}(1,0),\ O_1=I$ and $O_2$ is the permutation matrix for a transposition, we get $O_1^TAO_1+O_2^TAO_2=I$, which cannot possibly be equal to the singular matrix $B^{-1}AC$.